For $A=\begin{bmatrix}1&1\\0&1\end{bmatrix}$, start with $[A\ I]=\begin{bmatrix}1&1&1&0\\0&1&0&1\end{bmatrix}$. Apply $R_1\leftarrow R_1-R_2$ to obtain $\begin{bmatrix}1&0&1&-1\\0&1&0&1\end{bmatrix}$. Hence $A^{-1}=\begin{bmatrix}1&-1\\0&1\end{bmatrix}$.
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